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15 Ab Initio Molecular Dynamics 124 15 Ab Initio Molecular Dynamics 15 Ab Initio Molecular Dynamics 124 15 Ab Initio Molecular Dynamics 15.1 Why use ab initio methods in MD? Most importantly ab initio methods are used to generate the forces needed for a MD simulation. This can be done in two obvious ways: • Fitting of an empirical potential using results from ab initio calcula- tions. • Generating the forces directly from electronic structure calculations as the MD trajectory evolves. In order to produce analytical potentials, accurate descriptions of interac- tions between all involved types of atoms are necessary. Experimental data for these interactions is, however, rare. Still, even the best fixed potential has some drawbacks, most importantly: • Many different atom types included in a model give rise to a steeply rising number of interactions that need to be parametrized. • Changes in electronic structure (bonding pattern) are generally not accounted for. Thus, for the modelling of many phenomena, ab initio simulations are irreplaceable. Also, processes not foreseen (parametrized for) can take place – true predic- tive power. 15.2 What are ab initio calculations? 125 15.2 What are ab initio calculations? The term „ab initio” (Latin for „from the beginning”) is generally understood to denote methods that are devoid of experimental input; in other words derived from „first principles”. As we will see, many of the methods generally referred to as ab initio contain empirical parameters. Therefore the expression „electronic structure calcula- tions” is often more appropriate. In one way or the other, electronic structure calculations are all based on the Schrödinger equation @ ih¯ Φ(frig; fRI g; t) = HΦ(frig; fRI g; t) (15.1) @t This neglects • interactions between nucleons – almost always appropriate • relativistic effects. – problematic for heavy elements In principle, solution of the time-dependent Schrödinger equation descri- bes the dynamic behaviour of a system without any additional MD algorithm. To solve it is, however, almost always prohibitively expensive, therefore we must resort to different approximations. 15.3 Ab Initio MD There are three main approaches for combining electronic structure calcula- tions with molecular dynamics: • Born-Oppenheimer MD • Ehrenfest MD • Car-Parrinello MD There is a variety of codes available to perform ab initio MD (AIMD) si- mulations, e.g. ABINIT, CPMD, CASTEP, CP2K, NWChem, VASP. (not GROMACS, though. ) 15.3 Ab Initio MD 126 15.3.1 Born-Oppenheimer MD The Hamiltonian with electronic frig and nuclear fRI g degrees of freedom can be written as 2 2 X h¯ 2 X h¯ 2 H = − rI − ri 2MI 2me I i 2 2 2 ! 1 X e X e ZI e ZI ZJ + − + 4π"0 jri − rjj jRI − rij jRI − RJ j i<j I;i 2 X h¯ 2 = − rI + He(frig; fRI g) (15.2) 2MI I Here, He is assumed to be the Hamiltonian for the electronic system when the nuclei are stationary. Next, let’s assume that the exact solution of the corresponding time- independent electronic Schrödinger equation He(frig; fRI g)Ψk = Ek(fRI g)Ψk(frig; fRI g); (15.3) is known for all possible positions of the nuclei. Now it’s possible to expand the total wave function for the time-dependent Schrödinger equation as 1 X Φ(frig; fRI g; t) = Ψl(frig; fRI g)χl(fRI g; t) (15.4) l=0 with the nuclear and electronic wave functions normalized. The sum includes the complete set fΨlg of eigenfunctions for He, and the nuclear wave functions fχlg can be viewed to be time-dependent expansion coefficients. When inserted into the time-dependent Schrödinger equation, after mul- ∗ tiplication from the left by Ψk(frig; fRI g) and integration over electronic coordinates r, a set of coupled differential equations result " 2 # X h¯ 2 X @ − rI + Ek(fRI g) χk + Cklχkl = ih¯ χk: (15.5) 2MI @t I l Here Ckl are coupling operators, which depend on the kinetic energy and momenta of the nuclei. In adiabatic approximation the non-diagonal elements of Ckl are disre- garded. In Born-Oppenheimer approximation it is furthermore assumed that 15.3 Ab Initio MD 127 Ckk ≈ 0, which leads to: " 2 # X h¯ 2 @ − rI + Ek(fRI g) χk = ih¯ χk (15.6) 2MI @t I This approximation is safe for most physically interesting cases. The next step is to approximate the nuclei as classical point particles. This is done by rewriting the corresponding wave function in terms of an amplitude factor Ak and a phase Sk χk(fRI g; t) = Ak(fRI g; t) exp[iSk(fRI g; t)=h¯] (15.7) Using the following transformation for the momenta of the nuclei: PI ≡ rI Sk; (15.8) _ the Newtonian equations of motion PI = −∇I Uk(fRI g) can be written as dPI dt = −∇I Ek or (15.9) ¨ BO MI RI (t) = −∇I Uk (fRI (t)g) which holds separately for each decoupled electronic state k. • The nuclei move according to classical mechanics in an effective po- BO tential Uk , which is given by the Born-Oppenheimer potential energy surface Ek. • Ek is obtained by solving the time-independent electronic Schrödinger equation for the kth state at the given nuclear configuration fRl(t)g. • Because we now directly obtain the forces from the Born-Oppenheimer total energy Ek, this approach is often called Born-Oppenheimer mo- lecular dynamics. • Because the time-independent Schrödinger equation was utilized for the electronic system, this approach doesn’t maintain the quantum mecha- nical time evolution of the system. 15.3 Ab Initio MD 128 15.3.2 Ehrenfest MD Another approach, which does maintain the QM time evolution, involves di- rectly separating the total wave function Φ(frig; fRI g; t) so that the classical limit can be imposed for the nuclei only. The simplest form is a product Z t i ~ 0 0 Φ(frig; fRI g; t) ≈ Ψ(frig; t)χ(fRI g; t) exp Ee(t )dt ; (15.10) h¯ t0 where both wave functions again are separately normalized at every instant of time. To simplify equations, a phase factor Z ~ ∗ ∗ Ee = Ψ (frig; t)χ (fRI g; t)HeΨ(frig; t)χ(fRI g; t))drdR (15.11) is introduced. Inserting this wave function to the equations for the Hamiltonian and to the Schrödinger equation and implying conservation d hHi =dt ≡ 0 can be shown to lead to the following relations @Ψ X h¯ ih¯ = − r2Ψ @t 2m i i e Z ∗ + χ (fRI g; t)Un−e(frig; fRI g)χ(fRI g; t)dR Ψ (15.12) @χ X h¯ ih¯ = − r2χ @t 2M I i I Z ∗ + Ψ (frig; t)He(frig; fRI g)Ψ(frig; t)dr χ. (15.13) These coupled time-dependent Schrödinger equations form the basis of the time-dependent self-consistent field (TDSCF) method introduced in 1930 by Paul Dirac. Both the electrons and the nuclei move quantum mechanically in time-dependent effective potentials (i.e., average fields, given in the curly brackets). Next step is to again approximate the nuclei as classical point particles, which is done similarly to what was shown for the Born-Oppenheimer case. This gives for the equations of motion: dP Z I = −∇ Ψ∗H Ψdr or dt I e ¨ E MI RI (t) = −∇I Ue (fRI (t)g) (15.14) 15.3 Ab Initio MD 129 E Here Ue = hΨjHejΨi is now the Ehrenfest potential, which is given by quantum dynamics of the electrons by solving the time-dependent electro- nic Schrödinger equation • The AIMD approach using these equations of motion simultaneous- ly to the time-dependent Schrödinger equation for electrons is called Ehrenfest molecular dynamics. • This is a mixed approach where nuclei are treated as classical particles, but electrons retain their QM nature. • As a limiting case to the Ehrenfest method, it can be reduced to the Born-Oppenheimer method by restricting the electronic system to its ground state. 15.3.3 Comparison Between BOMD and EMD In the Born-Oppenheimer approach, the electronic state is obtained from the time-independent, i.e., stationary, Schrödinger equation. This implies that the time dependence of the electronic system is dictated by the motion of the nuclei, which it just follows. The method can be expressed for the electronic ground state as ¨ MI RI (t) = −∇I minΨ0 fhΨ0jHejΨ0ig E0Ψ0 = HeΨ0: (15.15) It is possible to consider also a certain excited state Ψk; k > 0 within the Born-Oppenheimer method, but without any interferences with other states. • The BOMD is very dependent on that a minimum hHei is reached in each time step to ensure good results. In the case of Ehrenfest MD, the forces are calculated „on-the-fly” as the nuclei are propagated using classical mechanics. They can be numerically solved simultaneously from the coupled set of quantum/classical equations Z ¨ ∗ MI RI (t) = −∇I Ψ HeΨdr = −∇I hHei (15.16) " # @Ψ X h¯2 ih¯ = − r2 + U (fr g; fR (t)g) Ψ @t 2m i n−e i I i e = HeΨ: (15.17) 15.3 Ab Initio MD 130 This can be done conveniently by expanding the electronic wave function Ψ as 1 X Ψ(frig; fRI g; t) = cl(t)Ψl(frig; fRI g); (15.18) l=0 where cl(t) are complex coefficients which determine the occupations of each state l and Ψl are solutions to the time-independent Schrödinger equation. This approach includes the transitions between different electronic states Ψk and Ψl within the framework of classical nuclear motion and the mean-field approximation of the coupled problem. • The Ehrenfest method includes electron dynamics and requires there- fore a very short time step. 15.3.4 Car-Parrinello MD Taking into account the strengths and weaknesses of BOMD and EMD, an ideal AIMD method would 1.
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